Stochastic volatility: Bayesian computation using automatic differentiation and the extended Kalman filter

Abstract

Summary Stochastic volatility (SV) models provide more realistic and flexible alternatives to ARCH-type models for describing time-varying volatility exhibited in many financial time series. They belong to the wide class of nonlinear state-space models. As classical parameter estimation for SV models is difficult due to the intractable form of the likelihood, Bayesian approaches using Markov chain Monte Carlo (MCMC) techniques for posterior computations have been suggested. In this paper, an efficient MCMC algorithm for posterior computation in SV models is presented. It is related to the integration sampler of Kim et al. (1998) but does not need an offset mixture of normals approximation to the likelihood. Instead, the extended Kaiman Filter is combined with the Laplace approximation to compute the likelihood function by integrating out all unknown system states. We make use of automatic differentiation in computing the posterior mode and in designing an efficient Metropolis-Hastings algorithm. We compare the new algorithm to the single-update Gibbs sampler and the integration sampler using a well-known time series of pound/dollar exchange rates.